Constructing the dual complex. These \(d\) -bubbles are key to defining a topology, in particular, a \(D\) -dimensional cellular complex structure. To this end, we shall construct the dual finite abstract simplicial complex. Quite clearly, the graph complex and the dual complex are the same topological space. Note that we write \({\mathcal {H}} \subset {\mathcal {G}}\) , if \({\mathcal {H}}\)  is a subgraph of \({\mathcal {G}}\) . To construct the dual complex [1]}:
